A variational approach to the magneto-elastic bucklingproblem of an arbitary number of superconducting beamsCitation for published version (APA):Lieshout, van, P. H., & Ven, van de, A. A. F. (1990). A variational approach to the magneto-elastic bucklingproblem of an arbitary number of superconducting beams. (RANA : reports on applied and numerical analysis;Vol. 9008). Eindhoven: Technische Universiteit Eindhoven.

Document status and date:Published: 01/01/1990

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Eindhoven University of TechnologyDepartment of Mathematics and Computing Science

RANA9D-08

July 1990

A VARIATIONAL APPROACH TO THE

MAGNETO-ELASTIC BUCKLING

PROBLEM OF AN ARBITRARY NUM­

BER OF SUPERCONDUCTING BEAMS

byP.H. van Lieshout

A.A.F. van de Yen

Reports on Applied and Numerical Analysis

Department of Mathematics and Computing Science

Eindhoven University ofTechnology

P.O. Box 513

5600 MB Eindhoven

The Netherlands

A variational approach to the magneto-elastic buckling problemof an arbitrary number of superconducting beams

P.R. van Lieshout andAAF. van de Yen,

Eindhoven University ofTechnology,Department ofMathematics and Computing Science,

P.O. box 513,5600 MB Eindhoven, The Netherlands,

t authorfor correspondence).

ABSTRACf

Based upon a variational principle and the associated theory derived in three precedingpapers, an expression for the magneto-elastic buckling value for a system of an arbitrarynumber of parallel superconducting beams is given. The total current is supposed to beequal both in magnitude and direction for all beams, and the cross-sections are circular. Theexpression for the buckling value is formulated more explicitly in terms of the so-calledbuckling amplitudes, the latter following from an algebraic eigenvalue problem. The per­tinent matrix is formulated in terms of complex functions, which are replaced by real poten­tials. The matrix elements are calculated by a numerical method, solving a set of integralequations with regular kemals. Besides the buckling value(s) also the buckling modes areobtained. Finally, our results are compared with the results of a mathematically less compli­cated theory, Le. the method of Biot and Savart.

where the intermediate (or rigid-body) fields B and A must be determined from (cf. [2], (1.7),(1.8»

The constraints for 'I' are so severe that, by given u, 'I' is completely determined by (1.4) (that iswhy we used the notation 1 =1 (u ;10) instead of1 =1 (u t '1'; 10»' •We now can propose the following variational approach to the magneto-elastic buckling problemof a superconducting structural system:

(1.1)

(1.4)

(1.3)

(1.2.1)

(1.2.2)

(1.2.3)

4~ER2 1 f [ v ]1(u;/o)=- 2 -1- -12 elc/cell+e/de/d dV~/o +V a- - v

+ f ['I' (Bj Ui.j - Bi.j Uj ) +Ble Blc,j Uj Uj - ejjm BmAj./d Ule U,dG

+2 B le (eijle UI-eljle Ui ) (Aj,m Um ),1+t B Ic B le (uj,j Uj -Ui,j Uj )] Nj dS

- f Tjle Ui,le Ui.j dV ,a-

whereas the perturbed magnetic potential 'I' is related to the displacement field u according to

.:1'1'=0 , xe G+; ~~ =(Bjui,j-Bj,juj)Nj t xe dG;

ejjIeBlc,j=O, (orAj,jj-Aj,jj=O) , xe G+;

B j Nj =0 , (or A= constant) , x e dG ;

B-.+c(x), I x 1-'+00 ;

and the pre-stresses Tjj have to satisfy

Tij,j=O, xe G-; TjjNj=-t (B,B)Nj , xe dG;

'1'-.+0 , I x 1-.+00 .

1. Introduction

In this paper the variational character of the method for the calculation of the magneto­

elastic buckling value for superconducting structural systems is shown to full advantage. This

variational method was derived in [1] and applied in [2] and [3] to pairs of superconducting

beams and rings, resPeCtively. Instead of using the explicit relations for the buckling values, as

[2], (1.6) and [3], (2.25), we here start anew with the formulation of a functional 1=1(u;/o)

(taken from [1]). In this, u is the displacement field (in buckling) and lois the total electric

current of the superconducting (slender) structure. This functional 1 is given by [1], (7.10). More­

over, we consider the relations [1], (7.12), (or [2], (1.7), (1.8» and [1], (7.15), (or [2], (1.9» as

constraints. Since this paper concerns systems of superconducting beams, we will use the normal­

ized variables as introduced in [2], (3.1). We then can derive from (7.10) (along the same lines as

[1], (7.18) is derived) the following expression for 1 (for the definition of the symbols we refer to[l], [2])

-2-

the displacement field u is derived from the variation of J with respect to u and, then, the

buckling value for the current lois obtained by putting J equal to zero (see [1], (2.14»;

hence, this means that we have to solve

The function c (x) in the relation (1.2.3) is characteristic for the problem under consideration, but

(after the normalization) independent of the current 10 (see e.g. [2], (3.3». This relation is made

more specific in Section 2, eq. (2.3). Therefore, the current 10 only turns up in the functional J

through the factor 4~ER2/1J<l15 in the first term of J (see (1.1», and so the buckling current can

indeed be calculated by (1.5)2. The approach to calculate u from (1.5)1 is different from that in

[2] and [3], where an a priori choice for u was made (however, based on rather trivial physical

arguments).

In the next section we shall apply the method described above to a system of an arbitrary

number N of slender superconducting beams, placed parallel to each other in one plane. We shall

choose the displacements of the respective beams out of a class of displacement fields represent­

ing the bending of a slender beam. The best member of this class is found by application of

(1.5)1• In this wayan eigenvalue problem for the amplitudes of the buckling displacements of the

beams is found. This eigenvalue problem is governed by a symmetric matrix A. The highest

eigenvalue of A corresponds to the lowest buckling value of 10 , For the calculation of the matrixA the fields B and 'I' are needed. The mean part of this paper is concerned with the calculation of

these fields. For N > 2 it seems no longer possible to find an analytical solution for B and 'I' (as in[2]) and, therefore, we have to set up a numerical procedure for this calculation. This procedure is

presented in Section 4. In Section 5 the numerical results are given. In the final section somespecific results are presented and a comparison with the so-called Biot-Savart-method (cf. [2],

[3]) is made. SH 2. A set ofN parallel beams

In [2], Section 4, the authors gave a detailed description of a system of two infinitely long

parallel slender beams. For the choice of the coordinate axes el, e2 and e3, we refer to Fig. I. We

restrict ourselves to beams having circular cross-sections, radius R (this is not necessary at thispoint, since the following analysis analogously holds for cross-sections which show double sym­

metry; cf [2], Section 4). The centers of the cross-sections all lie on the el-axis at distances 2a

from each other. The infinitely long beams are periodically supported over length I. We number

the N beams with n, ( 1~ n~N). The central line of the first beam coincides with the e3 -axis. The

regions occupied by the cross-sections in the el ~-plane are denoted by D;, (1~ n~N), with

boundaries'dD,u and the 2-dimensional vacuum space outside the beams is D+. The position ofthe center ofDII is x,,=2(n-l)a el.

(1.5)5..J=O , and J=O .

-3-

I D-/0-_; . _. _. _. _._. _ ._._._~-- ._.~.'~._

---- A 'oN 4 - - i aD.v

--- - ----- - - - -- - -- - ---- --,---- - - - - - - - - - - - - - - - - - - - D+ - --- - _ _ _ _ _ _ _ G+ - - + - -

2Na .I D-

/0-- - - ~. 2__ u-A; '-' -'-' -'- "Gi'-' ;A.-' -'l~~~aD~

~1 ~/0 -- - - Dr----.. Ar;\~.-. -·-l-"-\;T·-~-~- "02 i ;~-

~ .j

Fig.l. A set ofN parallel beams

d(xn =2(n-1)a ; '= dz ; lSnSN).

As in [2], (2.5), the problem (1.4) for the perturbed magnetic potential 'If is reduced to a 2­dimensional problem by the separation of variables

(the relationship between wn(z) and W (z) will be derived furtheron, see (2.7». The intermediate(or rigid-body) field B (subjected to the constraints (1.2» is already purely 2-dimensional, i.e.B=B (x ,y) and (B, e3 )=0. The condition at infinity, (1.2.3), is replaced by the set of condi­tions (compare with [3], (2.6» (or is the unit tangential vector along aDn )

B~O , x2+y2~oo ,

(2.2)

(2.1)

'If(x,y,z)=q.(X,y)w(Z) ,

In the sequel it is supposed that the total currents, running along the surfaces of the superconduet­

ing beams, are all equal both in magnitude (10) and in direction. In the undeformed state of thesystem the currents are in the positive e3-direction. Analogous to [2], (4.1), the displacement fieldu(n) (x), xeD;, of the n-th beam is expressed in terms of explicit functions of the in-plane vari­

ables x and y and the displacement W n (z ) of the central line, according to

u~n) (x ,y ,z)=wn(z)+t v[(x-xn )2_y 2]w:(z) ,

(n) ..U2 (x,y,z)=v(x-xn)ywn(z) ,

u~n)(x,y,z)=-(x-xn)w:(z), (x,y)eD;;

J(B,or)ds=27tR , lSnSN,cw.

(2.3)

where the last condition (i.e. Ampere's law in the normalized variables) expresses the relation

between the (normalized) rigid-body field B on the boundary aDn and the total current on the n-th

(2.4)

-4-

beam.The constraints (1.2) for the rigid-body field B=B%(x ,y )el +By(x ,y )e2' can now be written out

explicitly, yielding

aB% aBy aB% aBy--+-=0, -=-, (x,y)eD+;

ax dy ay ax

B%N%+ByNy=O, (x,y)e aDra ;

f (-B%Ny+ByN%)ds=2TCR , (l~n~N);aD.

(B%,By)~O , x2+y2~oo.

As concerns (1.3) we only note that, in accordance with the boundary condition (1.3)2 the nor­

malized pre-stresses Tij are of the order of B 2 = (B , B).

The constraints (1.4) for 'I' can be evaluated by substitution of (2.1) and (2.2) into them. In doingso we neglect tenns of order R21l 2• This means in practice, that we maintain in (2.1) only thezeroeth order tenn, i.e.

u~lt) =wra(z) , u~ra) =u~lt) =0 .

The boundary condition (1.4)2 thus becomes

~ acp(x,y) aB%(x,y)aN= aN w(z)=- aN wra(z),(x,y)eaDra .

Since this relation must be satisfied for arbitrary z, it is necessary that

wra(z)=vraw(z) , (vrae /R.,I~n~N).

(2.5)

(2.6)

(2.7)

We call the numbers Vra the amplitudes of the buckling displacements, and we note that the vra'sare independent of each other. Furthennore, the separation (2.2) is only then consistent with theLaplace equation (1.4)1 if there exists a parameter AE /R.+ such that

l\CP(X,y)-A2 cp(X,y)=O and W"(Z)+A2 W(Z)=0 . (2.8)

The parameter Ais related to 1through the support conditions of the beams (which are supposed

to be the same for all beams). For simply supported beams Aequals 1CI1.In this way the following constraint relations forcp(x,y) are obtained from (1.4)

l\CP=A2 cp, (x,y)eD+;

acp aB%aN=-vra aN ' (x,y)eaDII , (l~n~N); (2.9)

The amplitudes VII of the central line displacements and the buckling value for 10 are stillunknown and are to be solved from the variation c.q. zeroness of the functional J, i.e.

for each m e [I,N].

Finally we note that (as Tij is of the order B 2) the third integral gives a contribution that is ofo (R 21l2 ) and, hence, negligible Gust as was found in [2] and [3]). All this yields, apart from afactor

(2.13)

(2.12)

(2.11)

(2.10)

acjlmaN =0, (x ,y)e aD\aDm ;

-5-

aJ-a=0 (lSnSN), and J=O.

v"

NeIl(x,y)=1: vmell",(x,y) .

m=1

(which might be normalized to unity) the ultimate expression for the functional J, Le.

J=J(v;Io)=(A v, V)-K(V, v),

which is exact up to 0 (A,2 R 2 }( v, v). Here v is a N-vector, representing the buckling ampli­

tudes, which possesses the following column representation with regard to the orthonormal, posi­tively orientated base {E1 , ••• ,EN} of /RN,

V=[Vl''V2'''','VN]T; (2.14)

K is a positive scalar, which represents the entrance into the functional of the current 10 ,

4x2El A,4R 2

K= '2 ,I,=Jx2 dS=.!.1tR 4 , (2.15)~Io Dj 4

and A is a linear transfonnation from /RN~ /RN' having the following matrix with regard to the

We proceed with the evaluation of the expression for J according to (1.1) for the displacement

field (2.1). Firstly, we note that in the fonnula (1.1) for the functional J the regions G+, G- and

the boundary aG are to be restricted to the truncations D+ x [0, p ], D- x [0, p ] and aD x [0, p ],respectively, where D- and aD are the unions of the regions D; and the boundaries aD", respec­

tively. This is based upon the assumption that the fields are periodic in the z- or f3 -direction with

period p (see [2], section 2, for more details).The right-hand side of (1.1) contains three integrals. The first one, representing the elastic energy,

yields in the usual way the classical energy for a slender beam in bending (see [2], (2.2». Sincewe neglect terms of 0 (R 21l2 ) (or 0 p.2R 2 ), as A, is proportional to r 1) we may use in the ela­

boration of the second integral the reduced form (2.5) for the displacement field. Moreover weuse (2.2), (2.4)1.2, (2.7) and (2.8)2, and we introduce the set of functions eIlm' (IS mS N), by

Then (2.9) implies that each cjlm is independent of the amplitudes Vb V2, •••VN and has to satisfy

.1cjlm=A,2cjlm , (x,y)e D+ ;

acjl", aB%aN =- aN ,(x,y)e aD", ,

(2.17)

-6-

(2.16)

On account of the Helmholtz problem (2.12) and Green's second identity we derive from the

matrix representation formulas (2.16) the propeny

J d<ll/l J d<llmA_ = <11m a ds= <11/1 aN ds=AIII/I' n:#m .

aD N aD

Hence, the linear transformation A is symmetric and elaboration of (2.10) yields

(A V, v)A V=KV , v:#O, K> 0; K= ). (2.18)

(v,v

The set (2.18) implies that the lowest buckling value for the current 10 corresponds to the highestpositive eigenvalue K of the matrix A. This matrix still depends on the parameter A. by means ofthe functions <11m (cf. (2.12)1). In the next section we shall prove that for slender beams theinfluence of the ratio R/1 on the eigenvalue for K is negligible.

3. Complex formulation

In this section we shall use a great deal of the complex manipulations, which were alreadyapplied to the buckling problems for one single beam and for a set of two parallel beams in [2].

Therefore, we shall recapitulate only those notations and methods, which are indispensable to theunderstanding of the complete procedure. We introduce a small parameter ~ (0 <~ « 1), the nor­malized complex coordinate z and the complex function F in the same way as in [2], (2.7), (3.7),(3.25), Le.

~=AR , z=(x+i y )/R ,

(3.1)

where S+ and C stand for the region and curves in the complex z-plane corresponding to D+ andaD, respectively. Moreover~ we denote the z-transformations of D; and aD/I by S; and C/I,respectively.Analogous to [2], (3.26), (4.2), (4.4) the relations for the rigid-body state (see (2.4» can betransformed into (for the definition of the complex line element dz see [2], (3.22»

F analytical , z e S+ •

FdzeR, zeC.

F ~O. I z 1-+00 , (3.2)

-7 -

JFdz=21t , ISnSN .c.

The introduction of the real-valued functions (compare with [2], (3.28), (4.5) and note the differ­ence between the definition of1m used here and the one according to [2], (4.5»

, IS I Z-ZfI ISa/R, n*m,

(3.3)

What we are looking for are the numerical values of the coefficients Amra according to (3.4) and,hence, it is evident that our special interest is in the boundary values of the functions 1m. For thecalculation of these values an integral equation is consnueted. Since the consnuction runs alongthe lines of the methods presented in [2], (3.31)-(3.46) and (4.7)-(4.15) we do not enter intofurther details here, but only state the main results. Also, we use the convention that anyo (02 lot 0)-tenn is referred to as an 0 ( 02 )-tenn.The functions 1m are asymptotically approximated by the o-independent functions gm' accordingto

(3.6)

(3.7)

(3.5)

(3.4)

(3.8.1)

Im(z)=gm(z)(I+0(02», ZE C, ISmSN,

where gm satisfies (compare with [2], (4.10.2»

1 1 f gm (z)'2 gm(zo)+Re{-2. dz}=R(zo) ,

1U C z-zo

R(zo)=Re{ 21 . JF(z)dz} , ZOEC\Cm ,1U c. z-zo

with

(lS m,nS N) enables us to write (2.16) as (for the definition of the complex derivative (J/(Jz, see[2], (3.24»

J(JBx J dF

Amra =-2 1m 1m -(J- dz=-Im 1m dz dz,c. z c.

and (2.12)2 as

(JIm(IN =0 , ZE C, ISm,nSN.

and

1 J F (z)R(zo)=Re{~ --dz} , ZOE Cm ,

1U c'C. z-zo(3.8.2)

Cauchy's theorem for analytical functions states that

_l_JF(Z)dz=_l_J F(z)dz+_1_ J F(z)dz=2m c Z-Zo 21ti c Z-Zo 2m c'C z-zo

• •

-8-

=0 , Zo E S- •

=-F(zo) , ZOE S+.

Introduction of the N analytical functions (so-called Cauchy-integrals)

<1>m(Zo)=~Jg".(z)dz-~ JF(z)dz,zOEC\C,21tl c Z-Zo 21tl c. Z-Zo

and use of (3.9) in (3.7)-(3.8) leads us to the following set of Riemann-Hilbert problems

Re<1>;(zo)=O. ZOE C,

and

Im[<1>;(zo)-<1>:;'(zo)]=-ImF(zo) , ZOE C".,

=0 , ZOE C\C"..

Furthennore. the functions g". are related to the Cauchy-integrals <1>".,

g".(zo )=<1>;(zo )-<1>:;'(zo )+F (zo) • Zo E Cm ,

=<1>;(zo)-<1>:;'(zo) , ZOE C\C"..

(3.9)

(3.10)

(3.11.1)

(3.11.2)

(3.12)

Since <1>m is analytical in S- it follows from (3.11.1) that <1>; equals an imaginary constant, Le.

<1>;(Z )=i c"'" • ZE S; . CIM E IR . (3.13)

Substitution of (3.6), (3.12) and (3.13) into the expression for A"", according to (3.4) yields, under

the neglect of0 ( ri )-tenns.

J dF J dgmA"",=-Im gm-dz=Im F--dz=

c dz c dz• •

(3.14)

(3.15)

Using the short-hand notation

d<1>+F". (z ) = dzm • Z E S+ U C ,

we arrive at the ultimate mathematical fonnulation for the detennination of the buckling current

10:

Calculate the matrix A from

A"",=-Im JF Fmdz • 1Sm,nSN.c.

where the functions F ( Z ) and Fm( Z ) satisfy

(3.16)

-9-

F , Fm analytical , z e S+ ,

F , Fm -+0, I z 1-+00 •

JF dz=21t • JFm dz=O.c. c.

(3.17)

Im(Fdz)=O. ze C.

elFIm(Fmdz)=Im( dz dz). zeCm ,

=0 • z e C\Cm ;

and, then, the amplitude-vector v and the buckling current 10 are obtained from the eigenvalue

problem

A V=KV , v;tO • K>O, (3.18)

and the relation

In other words, the amplitudes of the central line displacements always cancel each other.

and as a consequence, the column-sums of the matrix A are equal to zero. Use of this property in(3.18) shows us that

(3.21)

(3.19)[

El ] 112I =21t02 yo ~KR2

4. Numerical procedure for the calculation of the matrix A

In [2], for the case of two circular rods, the region S+ was transformed into a ringshapedregion by conformal mapping and the resulting problem was solved by complex analysis. For thecase N > 2 such an analytical treatment is impossible and, therefore, we search for a numericalsolution procedure for the eigenvalue problem (3.18). This, more specifically, amounts in anumerical calculation of the elements AIM of the matrix A, according to (3.16).The first step is to reformulate the problem (3.16)-(3.20) in real terms, by introduction of the realfunctions Cll= Cll (x ,y ) and Cllm = Cllm (x , y ) through

On account of the fact that, within our approximation. the matrix A is independent of the parame­ter a, it is evident that the buckling current lois proportional to 02

• Moreover, we note that (3.17)directly implies that

N elF1: Fm =dz • ze S+u C, (3.20)m=1

(4.1)

-10 -

aro . aro arom . aromF=-~-l ~ ;Fm=-~-l~,

for 1~m~Nand x=(x,y)e S+ U C.The problem then transfonns into (with dz=i N ds =(iNx -Ny )ds, and aro/as =0 (see (4.3.3»):

Find the positive eigenvalues 1C of the matrix A with elements

where ro and COm satisfy

~ro=o, ~rom=O, xe S+ :

Vro~O, Vrom~O, I x I ~oo;

aro arom a aroas =0, xe C ; ---a;-=OIM as (NA: aN) , xe CII :

Jaro JaromaN ds=21t, aN ds=O,

c. c.

for 1~m,n~N.

(4.2)

(4.3.1)

(4.3.2)

(4.3.3)

(4.3.4)

With (4.3.1) and (4.3.4) the conditions at infinity (4.3.2) can be made more explicit, yielding

oo=Nlog I x 1+0(1), rom=O(1) , I x I ~oo. (4.4)

(4.5)

If wished for, the 0 (1 )-terms (constants) in (4.4) can be made zero, Le. replaced by 0 (1)­

terms, because the potentials ro and rom are only relevant up to a constant tenn.Moreover, the boundary conditions (4.3.3) can be integrated along each separate boundary CII ,

giving

arooo=an , rom =0_ Nz aN + ~11/11 , X e CII ,

where an and ~nm are constant factors, which shall be determined furtheron from (4.3.4).In the second step the functions ro and rom are split up in a set of hannonic functions (in S+ u C),

which are bounded at infinity and known on the boundary C, according to (xk=2(k-1)aIR el' thecenter of the k-th cross-section)

N N00= L log I X-Xk 1+'1'+ L ~"k •

1=1 1=1

(4.6)N

C1)m = 'I'm + L ~km"1 .1=1

The first term of (4.6) is chosen in such a way that the first condition of (4.3.4) is satisfied. The

functions 'I' and 'I'm have to satisfy the boundary conditions (4.5) with an =~11/11 =0. as the remain­

ing part of these boundary conditions are fulfilled by the parts with "/c. All the unknown functions

-11 -

(Le. '1'., 'I'm and u,,) can be found from an exterior Dirichlet problem, which general fonn reads(V=V(x,y»

.1 V=O , xe S+ ,

V=O(l) , Ixl~oo,

V=I , xe C ,

(4.7)

where lis a given function of x on the boundary C of the exterior region S+. In (4.7) we have to

read for V successively '1', 'I'm and u,,_ The associated boundary functions I are given by:

NforV='I', ~/(x)=-Lloglx-x"l,

"=1

0, xe C\Cm ,

for V = 'I'm , ~ I(x)=

{

o,xec\c",forV=u", ~ I(x)=

1, xe C" .

(4.8.1)

(4.8.2)

(4.8.3)

(4.10)

The coefficients a" and ~km are still to be detennined from (4.3.4). This results in the followingrelations (for ISm,nSN)

N f au" d'I'L a" aN ds=- f aN ds ,"=1 c. c.

(4.9)

It should be noted that the N relations of the set (4.9)1 and the N xN relations of (4.9)2 arelinearly dependent, because (for each m,k e [1 , N ])

f a'l' ds=Jd'l'm ds= f au" ar=O,caN c aN c aN

due to the fact that '1', 'I'm and Uk are hannonic in S+ and bounded at infinity. Therefore, in both ofthe sets of (4.9) one relation has to be dropped, which can be replaced by the following relationsat infinity

- 12-

NLllkUk+'V=O,lxl-+ oo ,k=1

(4.11)NL ~kmUk+'Vm=O, (lSmSN) , I x 1-+ 00 •

k=1

For the derivation of these relations it is necessary to replace in (4.4) the 0 (l )-symbols by

o ( 1)-symbols.With the use of (4.3.3) the expression (4.2) for Amn can be rewritten in the fonn

J aIDm a aID aIDAmn=c.[Nx aN -omnNya;(Nx aN)] aN ds . (4.12)

For the calculation of these integrals we first have to solve the basic problems (4.7)-(4.8). How­ever, from (4.12) we see that, practically, we are only interested in the values of the nonnal

derivatives along the boundaries, Le. aViaN for x E C,., IS nS N.The further procedure could be based on the use of layer potentials (cf. [4], [5]). However, intro­

duction of a simple layer potential for the function V leads us to a situation in which it is difficultto detennine the limit of V at infinity, and, moreover, the problem now involves a Fredholm

integral equation of the first kind (weakly singular), Le. an ill-posed problem for the density ofthe potential. On the other hand, introducing a double layer potential we arrive at a Fredholmintegral equation of the second kind, which in general is singular.To avoid these complications, we separate from V particular logarithmic solutions of the Laplaceequation. The remaining part of V can then be expressed in double layer potentials, the densitiesof which satisfy ordinary integral equations. This separation is of the following fonn

V(X)=V1(X)+V2(X) , XE S+uC,

where firstly

V1(x)=- 2~ JIl(Y) a~y log Ix-y I dSy, XE S+uS- ,

with Il ( x) satisfying

~ JJ.(X)--21 J JJ.(Y) jN

alog I x-Y I dsy=!(x), XE C,. ,

1t CIC. a 'Y

or in short-hand notation

L+{Il(x)}=!(X) , xe C,., lSnSN.

Secondly

for xe S+ U (S-\{Xl ,X2, .. , • XN}), where

(4.13)

(4.14)

(4.15)

(4.16)

(4.17)

- 13-

As we shall show furtheron, the numbers co' C1, ... CN can be chosen in such a way that V =fon C. Note that the integral equations (4.15) (or (4.16» and (4.20) possess indeed regular kemals.

Moreover, the nonnal derivatives of the double layer potentials V 1 and Vi are continuous acrossthe boundaries CII (see [6], p. 170), so (since ~ VI =0 and ~Vi =0, xeS;;)

aVl avi

J aN ds=O , J aN ds=O , IS I,nS N , (4.25)c. c.

V/(x)= __I_ J~/(y) -f-log I x-y I dsy , xe S+ U S- ,21t C aNy

while ~(x) has to satisfy

1L+ (~/(x)}= 21t log I X-XI I , xe CII •

Evidently

~Vl=O,xeSuS-,

~V2=0, xeS+;~V2=C/8D(X-Xl)' xeST,

(8D is Dirac's delta function) and

V1-+O, V2-+co=O(I) , Ixl-+ oo ,

where the latter is a consequence of (4.18). From (4.23) together with (4.13) it follows that

co=V_= lim V(x).1](1-+-

and then (from (4.17»

aV2 aVclI = I aN ds= JaN ds, IS nS N .

c. c.

(4.18)

(4.19)

(4.20)

(4.21)

(4.22)

(4.23)

(4.24)

(4.26)

(4.27)

Taking in (4.14) for V1(x) the exterior limit forx-+CII , denoted by V!(x), we arrive at (cf. [4],

p. 382; f stands for the principal value)

V!(x)=t ~(x)- 2~ t~(y) aty

log I x-Y I dsy

1 f a=f(x)- 21t c. ~(y) aNy log Ix-y I dsy ,xe CII ,

where the last step follows immediately from (4.15). Writing for y and for x e CII

y=(xlI+rcos~)el+rsin~~,

and

- 14-

x=(XIi +cos9 )el +singe2 ,

respectively, we find for y e Cli (Ny =cosq> el + sinl\> e2)

a [ a ] [(-X+Y,Ny )]aN log I x - YI = ar log I x - Y I = x _ 2

y r=1 I Y I r=l

= 1-cos( 9-I\> ) _ 1 (4 28)2(l-cos(9-ep» -2" . .

With (4.28) the integral on the right-hand side of (4.27) can be evaluated to (forxe CIl)

1 f a -21t c. 1l (y) dNy log I x-y I dsy =t J.1n , (4.29)

where Jl.1l stands for the mean value ofll on CIl' Le.

As a consequence of (4.29), (4.27) reduces to

1 ­V!ex)=!(x)-z J.1n , xe CIl .

In a similar way one deduces

N -mV!(x)=co+t L Cmllll , xe CIl ,

m=1

where ~: is the mean value of the density Il'" on Cli' The boundary condition

v (x)=Vt'ex)+V!(x)=!(x),

now yields

N -m -t L cmllll +co=t Iln , lSnSN .

m=1

(4.30)

(4.31)

(4.32)

(4.33)

This set, together with the relation (4.18), which is the necessary condition for the boundednessof V2( x) at infinity, constitute the basic set for the calculation of co' C1, .•• , CN (after Il and III

are known). We can write this total set in a more concise notation by introducing the N-columnvectors a and e and the (N xN )-matrix B by

1 - 1 -mall=- Illl , ell = 1 , B_ =- Illl ,

2 2

for 1S m,nS N. Then, the above mentioned set can be written as

(4.34)

(4.35)

In this system of linear equations the vector (cT , cO)T represents the unknown variables. Thevector e is a fixed one, whereas the matrix B and the vector a are known once the ordinary

- 15 -

integral equations (4.16) and (4.20) are solved (recall that this must be done for allfs out of thethree distinct sets presented in (4.8». Note also that for the solution of (4.35) we do not need to

calculate the functions V (x) or VI (x) and V2(x); these are only auxiliary functions.As a matter of fact we are only interested in the values of the nonna! derivative of V at the boun­daries Cn. For this purpose we consider the function

hence,

1 NV3(x)=V(x)--2 1: cmlog I X-Xm I,

1t m=l

NV3(X)=V1(x)+co-1: Cm Vm .

m=l

(4.36)

(4.37)

(4.38)

From the foregoing analysis it then follows that V3(x) is hannonic in S+, bounded at infinity andsuch that (from (4.25»

IOV3aN ds=O, ISnSN.

c.

These features guarantee the existence of a hannonic function W (X), x E S+ U C, the conjugatefunction of V3, such that

&W=O,XES+,

W=O(l) , Ixl~oo,

oW oV3 2l. IoNaN =--':\-= ':\ --2 :I L cmlog I X-XmI ,XE C ,

uS uS 1t oS m=l

(4.39)

(4.41)

since V=f, forxE C.The above problem for W is, apart from a irrelevant constant, uniquely solved by writing W as asimple layer potential, the density of which satisfies an ordinary integral equation with regularkernal. Thus (cf. [4])

W(X)=--21

Iv(y)log Ix-y I dsy , (4.40)1t c

with v following from

1 1 I a oV3-'2 v(x)- 21t C'C. v(y) oN

zlog Ix-y IdSy=----a;-' XE Cn ,

with av3las as given by (4.39)3. Since W is the conjugate of V3, the nonnal derivative of V3 onC equals the tangential derivative ofW along C, so

aVoW IoNaN = as + 21t aN m~l em log I x-xm I . (4.42)

According to (4.40)

(4.44)

(4.43)

(4.45)

-16 -

aw =__1 fv( ) (x-y,s%) das 21t C Y I x _ y 12 Sy

1 (x-y,s%) 1 J (x-y,s%)=-- Jv(y) 2 dSy-- [v(y)-v(x)] 2 dsy21t CIC. I x-y 1 21t C. I x-Y I

f(X-y,s%)

-vex) 2 dsy , xe C", lSnSN.C. I X-Y I

Analogous to (4.28) it can be shown that

(x-y, s%) = sine a-ell)I x-Y 1

2 2(l-cos(a-~»

which is an odd function of ~ around (e +1t), and, hence, the last integral in the right-hand side of

(4.43) is equal to zero. Thus we obtain from (4.42)-(4.43)

av 1 aNI (x - Y, s% )-=--- L cmlog 1X-Xm 1-- Jv(y) dSyaN 21t aN m==1 21t CIC. I x-y 12

J(X-y,~)

- (V(y)-v(X)] 2 dsy , xe C" .C. I X-Y I

When v (x) is known, i.e. solved from (4.41), aVIaN can be calculated from (4.45).Before proceeding with the explicit numerical calculations that will be presented in the next sec­

tion, we recapitulate here the main steps in the calculation of Amn• This procedure is built up in

three parts. namely (for V = '1', V = Uk, and V = 'I'm' respectively)

Part 1: V='I'.

i) Calculate l-1(x) from (4.16) with/ex) according to (4.8.1).

ii) Calculate J,l!(x) from (4.20) (note that this relation and, hence, also 1-1', is identical for each

V).

iii) Detennine a and B from their definitions (i.e. (4.30), (4.34» and solve (4.35) for (cT ,Co l;this also yields '1'( 00 )=co (see (4.24».

iv) Calculate vex) from (4.4l)together with (4.39)3.

v) Find d\jflaN from (4.45).

Pan 2: V=Uk, lSkSN.

i)-v) Analogous to Part I, only with/ex) from (4.8.3), whereas in iii) and v) Uk( 00) and aUklaN,. respectively, are obtained

vi) Calculate (Ii from (4.9)1 and (4.11)1.

vii) Find aCJ)/aN from (4.6)1.

Pan 3: V='I'm, lSmSN.

i) Use the result from Part 2 vii) to obtain/ex) from (4.8.2), and calculate l-1(x) from (4.16).

- 17-

ii) Take IJ.I(x) from Part 1 ii)

iii) Solve (cT , Co l analogous to Part 1 iii), which also gives "'m( 00 )=co.

iv) Calculate v(x) from (4.41) and (4.39)3.

v) Find d"'m1dN from (4.45).

vi) Calculate ~mn from (4.9i and (4.11)2.

vii) Find dromldN from (4.6)2.

The final step is then:

Use the results of Part 2 vii) and Part 3 vii) for the calculation of Amn (lS m,nS N) from(4.12).

In our numerical program we follow the calculation scheme recapitulated at the end of Section 4,but we compute .the matrix elements A_ for m < n only; the remaining ones follow from theidentities

5. Numerical evaluation and results

In the preceding section we described a procedure for the solutions of the exterior Dirichletproblem in two dimensions, especially directed towards the calculation of the nonnal derivativesof the magnetic potentials on the boundaries. In this procedure the Dirichlet problem was refor­mulated in tenns of integral equations. In our numerical program all occurring integral equationsare approximated by systems of linear algebraic equations by means of discretization. For theapproximations of the integrals and of the tangential derivative of V 3 we use trapezoidal rolesand central differences, respectively. The integrand of the last tenn on the right-hand side of(4.45) in case y -+ x equals av IdS, and, again, the latter is approximated by a central difference.The discretization is accomplished by dividing the circles C1, .•• , CN in M segments, eachwith angle h = 2'1tlM. The x- and y-coordinates of the associated nodal points are consecutivelynumbered as

X(k-l)M+j =[2(k-l)a + cosU -1)h] el + [sinU -1)h] e2 ,

for x e Ck, and

y(I-I)M+j = [2(1-1)a + cosU -1)h] el + [sin(j -1)h] ~ ,

for y e C1 , with

k.le [l.N],je [1.M],andh=~.

NL A_ = 0 • and A_ =AMI •

11-1

(5.1)

(5.2)

(see (3.20-21) and (2.17». Standard routines. such as the partial pivoting process. are used for thesolution of the obtained linair systems and for the calculation of the eigenvalues and eigenvectorsof an N x N-matrix. As a check for the accuracy of our numerical procedure we compare ourresults for N = 2 with those obtained earlier in [2]. Our results for KI 'It correspond to the values ofQs in [2], Table 4. 'The results forK/'It. obtained for M =40. and for Qs are listed in Table 1. We

- 18-

conclude that a very close agreement between lCIx and Qs exists.

Table 1. Values of lC/x for N = 2 and M = 40 and of Qs (from [2], Table 4) for various values of

aIR

aIR 1.5 2 3 4 6 8 10

lC/x 0.2205 0.1678 0.09328 0.05661 0.02653 0.01520 0.009810

Qs 0.220 0.168 0.0935 0.0568 0.0266 0.0153 0.00985

For N = 2, the first buckling mode (corresponding to the lowest buckling value or largest eigen­

value lC) is found to be

again in accordance with the results of [2].

Of course, also the eigenvalue lC =0 appears, with buckling mode

v = [.!. ..J2, 1... ..J2f ,2 2

(5.3)

(SA)

for A is singular. However, this eigenvalue has no practical relevance, because it yields aninfinitely high buckling current. The same phenomenon arises for N > 2. Therefore, in the sequel

the eigenvalue lC =0 is left out of consideration.

In the following Tables 2, 3 and 4 one finds the numerical results for the eigenvalue lC

(related to the buckling current according to (3.19)) and the eigenvector (or buckling modes) forN =3 , 4 and 5, respectively; here we have used M =40 and a I R =3.

Table 2. The eigenvalues and buckling modes for N = 3 and a I R =3, computed for M =40.

'1C/1t VI V2 v3

0.1393 -00408 0.816 -00408

0.0724 0.707 0 -0.707

Table 3.The eigenvalues and buckling modes for N =4 and a I R =3, computed for M =40.

lC/1t VI V2 V3 v4

0.1640 -0.238 0.666 -0.666 0.238

0.1183 0.500 -0.500 -0.500 0.500

0.0592 -0.666 -0.238 0.238 0.666

- 19-

Table 4. The eigenvalues and buckling modes for N = 5 and a I R = 3, computed for M = 40.

'K/rc vI v2 v3 v4 Vs0.1790 -0.144 0.490 -0.692 0.490 0.144

0.1459 0.335 -0.623 0 0.623 -0.335

0.1028 -0.528 0.245 0.566 0.245 -0.528

0.0501 -0.623 -0.335 0 0.335 0.623

The values for the buckling current 10, associated with the computed highest values of 'K,

can be obtained from (3.19). With

(5.5)

(5.6)

(5.7)

for circular cross-sections, and with

rcRo=A.R=-,

I

for simply supported rods, (3.19) yields

10=_1- ~ R3

_ rr ...J'KI rc 12 'J;;

With use of this fonnula we have compared the results for 3, 4 and 5 rods with the buckling

current for a set of2 rods. The results are listed in Table 5.

TableS.

(6.1)

5

0.722

4

0.754

3

0.818

N

6. Discussion

In [2] and [3], as an alternative way, a more technical approach to the solution of bucklingproblems for (super)conducting structural systems was discussed. The method is based upon ageneralization of the law of Biot and Savart (cf. [7], Sect 2.6). In [2] this method was applied to

the problem of two parallel rods. In a straightforward derivation, completely analogous to that of

[2]. this method can be generalized to systems of more than 2 rods. For instance. for three rodsthe following equations are obtained

Ely v1V (z)= k i (VI-V2)+.!.kI (VI-V3).4

E Iy v~ (z) = k i (2vz -v I -V3),

E Iy v~(z)= k 1 (v3-vz)+.!.kI (V3-Vl)'4

with

- 20-

fJ{) 15k 1=-­

8M 2

Under the boundary conditions

Vi (0) = V;'(0) = vi(l) = v7(1), i = 1,2,3,

the lowest eigenvalue of (6.1) is

1t4 E I

k l = 4"31

(6.2)

(6.3)

(6.4)

(6.5)

associated with the buckling mode

v,(z)=v,(z)=-t v,(z). v,(z)=A Sin[ n/ ] .

This buckling mode is identical to the first one ofTable 2.From (6.4) with (6.2)1 the following formula for the buckling current is found (with I, = 1t R 4 /4)

/0 = - (2 rC3 a R2

_ rr .'1"3 12 ,,-;;

Let us compare this results with (5.7). For a / R = 3 we obtain from (5.7)

~R3 _~10 =2.679 -12- ,,-;; ,

and from (6.5)

~R3 {!10 =2.449 -2- -.1 fJ{)

(6.6)

(6.7)

(6.8)

We see that the buckling value found by the Biot-Savart method is about 8% lower then the value

from the variational method. The same difference was also found in [2] for the set of two rods.

For the system of 5 rods. the Biot-Savart method yields the buckling mode

V2 =V4 =-0.72 V3, VI =Vs =0.22 v3 ,

which differs only slightly from the first buckling mode from Table 4, where

v2 = V4 =-0.708 V3, VI =VS = 0.208 V3

For the buckling current we obtained

/0=0.723 ~ ;2R2 {! ,yielding, for a / R = 3,

(6.9)

(6.10)

(6.11)

'If R 3 _ fT10 = 2.168 -12- 'J~ .

On the other hand, (5.7) gives for a I R =3

'If R3 {f10 =2.364 -2- -,

1 IJ()

- 21 -

(6.12)

(6.13)

and again a difference of about 8% is observed. Hence, we conclude that this relative difference is

independent of the number N.

Finally, we also calculated by the Biot-Savart method the buckling current for an infinite setof parallel rods. The result was that the buckling modes were related to each other by

Vj+l=-Vj' j=1,2, '"

while the buckling current was found to be

(6.14)

(6.15)

(6.16)

1 ='fi2aR2 _fE.

o 212 'J;;It is striking to note that this value for the infinite set is exactly a factor (1t/2) lower than thevalue for the set of two rods, which according to [2], (5.31) is equal to

10= 'lfaR2

_ fT .12 'J~

We proceed with the analogous version of Table 5, but now with the results from the Biot-Savartmethod.

Table 6. Ratio's of the buckling currents for N rods and for 2 rods, calculated by means of the

Biot-Savart method.

N

(Io)NI (/oh

3

0.816

5

0.723

00

0.637

(6.17)

(6.18)

We note that the above ratio's are independent of the value of a I R. Moreover, the differences in

the ratio's according to Table 5 and to Table 6 (for N = 3 of 5) are negligible. Hence, we maywrite (the subindices V and BS denote values according to the variational method and the Biot­Savan method, respectively)

[ ~ON ] = [~ON ] = qN (N) ,02 v 02 BS

where qN depends only onN and not on aIR. With the use of [2], (5.17), this relation implies that

qN ~R2 *(ION)v = -- -- -...JQs 12

IJ()

If we assume this relation of general validness (i.e. for all values of aIR and N) we can

- 22-

extrapolate the results of Table 5 for N = 3 and N = 5 to other values of aIR. To this end we usethe lI..JQs -values as given in [2], Table 4, for several values of aIR. Furthennore, we can alsofind a corresponding value for the infinite system. In this way we find for the coefficient i 0

defined by

the relation

. qN . (a N)lO=-- = lO -, .

..JQs R

(6.19)

(6.20)

Values for this nonnalized buckling current are listed in Table 7.

Table 7. Values of the nonnalized buckling current io found by extrapolation from the Biot­Savan results.

N

aIR 3 5 00

4 3.429 3.037 2.6746 5.005 4.432 3.9038 6.606 5.850 5.150

10 8.230 7.288 6.417

In conclusion, we state that we have found here a simple algorithm to extrapolate from the Biot­Savan results the, more exact but also much harder to obtain. buckling values as they should befound by the variational method. Due to the striking correspondence between systems of rodsand (parallel) rings, as found in [3], it may be expected that this result can be generalized to sys­tems of N (N~ 2) rings. This will enable us to apply a combined method (based partially upon avariational approach and partially on Biot-Savart like calculations) to more complex systemssuch as, for instance, helical or spiral shaped conductors (cf.[8]).

References ,[1] P.H. van Lieshout, PM.J. Rongen and A.A.F. van de Yen, A variational principle for

magneto-elastic buckling, J. Eng. Math. 21 (1987) 227-252

{2] P.H. van Lieshout, P.M.J. Rongen and A.A.F. van de Yen, A variational approach tomagneto-elastic buckling problems for systems of ferromagnetic or superconducting beams,J. Eng. Math. 22 (1988) 143-176

[3] P.R.JM. Smits, P.R. van Lieshout and A.A.F. van de Yen, A variational approach tomagneto-elastic buckling problems for systems of superconducting beams, J. Eng. Math. 23(1989) 157-186

- 23-

(4] S.G. Mikhlin, An advanced course of mathematical physics, North-Holland Publ. Co.,Amsterdam, London (1970)

[5] M.A. Jaswon and G.T. Symm,/ntegral equation methods in potential theory and elastostat­ics, Academic Press, London (1977)

(6] a.D. Kellogg, Foundations ofPotential Theory, Dover Publ., New York (1929)

[7] F.C. Moon, Magneto Solid Mechanics, John Wiley & Sons, New York (1984)

(8] A.A.F. van de Yen and P.H. van Lieshout, Buckling of superconducting structures underprescribed current, (to appear in) Proceedings of the IUTAM - Symposium on the Mechani-

cal Modellings ofNew Electromagnetic Materials, Hsieh (ed.) Stockholm (1990), pp ,Elsevier Science Pub., Amsterdam (19..)

/

PREVIOUS PUBLICATIONS IN THIS SERIES:

Number

89-29

90-01

90-02

90-03

Author(s)

M.E. Kramer

A. Reusken

J. deGraaf

A.AF. van de YenP.H. van Lieshout

Title

A generalised multiple shooting method

Steplength optimization and linear mul­tigrid methods

On approximation of (slowly growing)analytic functions by special polynomials

Buckling of superconducting structuresunder prescribed current

Month

December'89

January '90

February '90

February '90

90-04

90-05

90-06

90-07

90-08

F.J.L. Martens

J. de Graaf

Y. ShindoK. HoriguchiA.AF. van de Yen

M. KuipersAAF. van de Yen

P.H. van LieshoutA.A.F. vande Yen

A representation of GL(q, JR) in April '90L2(Sq-l)

Skew-Hennitean representations of Lie April '90algebras of vectorfields on the unit-sphere

Bending of a magnetically saturated plate April '90with a crack in a unifonn magnetic field

Unilateral contact of a springboard and a July'90fulcrum

A variational approach to the magneto- July '90elastic buckling problem of an arbitrarynumber of superconducting beams